We study the heat semigroup generated by two-dimensional
Schrödinger operators with compactly supported magnetic field. We
show that if the field is radial, then the large time behavior of
the associated heat kernel is determined by its total flux. An exact
formula for the heat kernel, and for its large time asymptotic, is
derived in the case of the Aharonov-Bohm magnetic field. We also
establish some on-diagonal heat kernel estimates and discuss their
applications for solutions to the heat equation.